Methods and apparatuses for in-phase and quadrature-phase imbalance compensation

ABSTRACT

At least one example embodiments discloses a method of compensating for in-phase and quadrature (IQ) imbalance in a base station. The method includes generating, at the base station, compensation filter weights based on a plurality of IQ imbalanced training signals, the generating including, determining the compensation filter weights based on the plurality of imbalanced training signals in a frequency domain. The method further includes filtering based on the compensation filter weights.

BACKGROUND

In transmitters and receivers, in-phase (I) and quadrature-phase (Q)phase signals become imbalanced due to analog circuits within atransmitter or receiver. For example, analog circuits may includedemodulators, low pass filters and operational amplifiers. An analogcircuit in a receiving path may have a zero intermediate frequency (IF)design that produces imbalanced I and Q signals.

IQ imbalance may be caused by a phase offset in demodulator cosine andsine signals, a gain difference between an I-path and a Q-path in thelow pass filters and operational amplifiers, and a path length (delay)difference between the I-path and Q-path.

For example, quadrature carriers in an analog modulator do not haveexactly the same amplitudes and an exact phase difference of 90 degrees.This causes cross-talk between the I and Q paths, which is referred toas IQ imbalance.

The nature of the IQ imbalance is that an amount of the I signal spillsinto the Q signal and an amount of the Q signal spells into the Isignal. The frequency response of the IQ imbalanced signal has not onlythe original signal but also an image, at the negative frequency. Forexample, a 5 MHz tone has an image signal at −5 MHz, with loweramplitude. A wideband signal has a wideband image with negativefrequencies. If the frequency band of the wideband signal overlaps withits image, the wideband signal and images are superimposed in theoverlapped frequencies.

SUMMARY

According to at least one example embodiment, IQ imbalance is correctedby digital filters using an algorithm according to at least one exampleembodiment to estimate weights of the digital filters.

The algorithm determines a set of filter weights (and length) to reduce(cancel) the IQ imbalanced images. The algorithm uses a frequency-domainleast-squares fit method. Tone signals are sampled and input to ananalog circuit of a demodulator or modulator and, thus, the outputtedtone signals are imbalanced. Outputted tone samples for each IQimbalanced tone signal, at different frequencies, are captured. Thealgorithm determines the DFTs (Discrete Fourier Transform) of eachsampled tone signal. The filter weights are adjusted to reproduce thetone and to eliminate the image (target value of 0). The filter weightsare determined based on a least-squares fit on the DFT values(frequency-domain values) of the imbalanced sampled tones.

At least one example embodiment discloses a method of compensating forin-phase and quadrature (IQ) imbalance in a base station. The methodincludes generating, at the base station, compensation filter weightsbased on a plurality of IQ unbalanced training signals, the generatingincluding, determining the compensation filter weights based on theplurality of imbalanced training signals in a frequency domain. Themethod further includes filtering based on the compensation filterweights.

At least another example embodiment discloses a base station configuredto generate compensation filter weights based on an plurality of IQimbalanced training signals. The generating includes determining thecompensation filter weights based on the plurality of imbalancedtraining signals in a frequency domain.

At least another example embodiment discloses a user equipment (UE)configured to receive a compensated signal from a base station, thesignal being compensated based on an plurality of IQ imbalanced trainingsignals and compensation filter weights based on the plurality ofunbalanced training signals in a frequency domain.

BRIEF DESCRIPTION OF THE DRAWINGS

Example embodiments will be more clearly understood from the followingdetailed description taken in conjunction with the accompanyingdrawings. FIGS. 1-4B represent non-limiting, example embodiments asdescribed herein.

FIGS. 1A-1C illustrate a base station according to at least one exampleembodiment;

FIGS. 2A-2D illustrate channel models according to at least one exampleembodiment;

FIG. 3 illustrates an IQ imbalance and compensation model according toat least one example embodiment; and

FIGS. 4A-4B illustrate a IQ compensation model according to at least oneexample embodiment.

It should be noted that these Figures are intended to illustrate thegeneral characteristics of methods, structure and/or materials utilizedin certain example embodiments and to supplement the written descriptionprovided below. These drawings are not, however, to scale and may notprecisely reflect the precise structural or performance characteristicsof any given embodiment, and should not be interpreted as defining orlimiting the range of values or properties encompassed by exampleembodiments. For example, the relative thicknesses and positioning ofregions and/or structural elements may be reduced or exaggerated forclarity. The use of similar or identical reference numbers in thevarious drawings is intended to indicate the presence of a similar oridentical element or feature.

DETAILED DESCRIPTION

Various example embodiments will now be described more fully withreference to the accompanying drawings in which some example embodimentsare illustrated.

Accordingly, while example embodiments are capable of variousmodifications and alternative forms, embodiments thereof are shown byway of example in the drawings and will herein be described in detail.It should be understood, however, that there is no intent to limitexample embodiments to the particular forms disclosed, but on thecontrary, example embodiments are to cover all modifications,equivalents, and alternatives falling within the scope of the claims.Like numbers refer to like elements throughout the description of thefigures.

It will be understood that, although the terms first, second, etc. maybe used herein to describe various elements, these elements should notbe limited by these terms. These terms are only used to distinguish oneelement from another. For example, a first element could be termed asecond element, and, similarly, a second element could be termed a firstelement, without departing from the scope of example embodiments. Asused herein, the term “and/or” includes any and all combinations of oneor more of the associated listed items.

It will be understood that when an element is referred to as being“connected” or “coupled” to another element, it can be directlyconnected or coupled to the other element or intervening elements may bepresent. In contrast, when an element is referred to as being “directlyconnected” or “directly coupled” to another element, there are nointervening elements present. Other words used to describe therelationship between elements should be interpreted in a like fashion(e.g., “between” versus “directly between,” “adjacent” versus “directlyadjacent,” etc.).

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of exampleembodiments. As used herein, the singular forms “a,” “an” and “the” areintended to include the plural forms as well, unless the context clearlyindicates otherwise. It will be further understood that the terms“comprises,” “comprising,” “includes” and/or “including,” when usedherein, specify the presence of stated features, integers, steps,operations, elements and/or components, but do not preclude the presenceor addition of one or more other features, integers, steps, operations,elements, components and/or groups thereof.

It should also be noted that in some alternative implementations, thefunctions/acts noted may occur out of the order noted in the figures.For example, two figures shown in succession may in fact be executedsubstantially concurrently or may sometimes be executed in the reverseorder, depending upon the functionality/acts involved.

Unless otherwise defined, all turns (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which example embodiments belong. Itwill be further understood that terms, e.g., those defined in commonlyused dictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art andwill not be interpreted in an idealized or overly formal sense unlessexpressly so defined herein.

Portions of example embodiments and corresponding detailed descriptionare presented in terms of software, or algorithms and symbolicrepresentations of operation on data bits within a computer memory.These descriptions and representations are the ones by which those ofordinary skill in the art effectively convey the substance of their workto others of ordinary skill in the art. An algorithm, as the term isused here, and as it is used generally, is conceived to be aself-consistent sequence of steps leading to a desired result. The stepsare those requiring physical manipulations of physical quantities.Usually, though not necessarily, these quantities take the form ofoptical, electrical, or magnetic signals capable of being stored,transferred, combined, compared, and otherwise manipulated. It hasproven convenient at times, principally for reasons of common usage, torefer to these signals as bits, values, elements, symbols, characters,terms, numbers, or the like.

In the following description, illustrative embodiments will be describedwith reference to acts and symbolic representations of operations (e.g.,in the form of flowcharts) that may be implemented as program modules orfunctional processes including routines, programs, objects, components,data structures, etc., that perform particular tasks or implementparticular abstract data types and may be implemented using existinghardware at existing network elements or control nodes (e.g., ascheduler located at a cell site, base station or Node B). Such existinghardware may include one or more Central Processing Units (CPUs),digital signal processors (DSPs),application-specific-integrated-circuits, field programmable gate arrays(FPGAs) computers or the like.

It should be borne in mind, however, that all of these and similar termsare to be associated with the appropriate physical quantities and aremerely convenient labels applied to these quantities. Unlessspecifically stated otherwise, or as is apparent from the discussion,terms such as “processing” or “computing” or “calculating” or“determining” or “displaying” or the like, refer to the action andprocesses of a computer system, or similar electronic computing device,that manipulates and transforms data represented as physical, electronicquantities within the computer system's registers and memories intoother data similarly represented as physical quantities within thecomputer system memories or registers or other such information storage,transmission or display devices.

Note also that the software implemented aspects of example embodimentsare typically encoded on some form of tangible (or recording) storagemedium or implemented over some type of transmission medium. Thetangible storage medium may be magnetic (e.g., a floppy disk or a harddrive) or optical (e.g., a compact disk read only memory, or “CD ROM”),and may be read only or random access. Similarly, the transmissionmedium may be twisted wire pairs, coaxial cable, optical fiber, or someother suitable transmission medium known to the art. Example embodimentsare not limited by these aspects of any given implementation.

As used herein, the term “user equipment” (UE) may be synonymous to amobile user, mobile station, mobile terminal, user, subscriber, wirelessterminal and/or remote station and may describe a remote user ofwireless resources in a wireless communication network. The term “basestation” may be understood as a one or more cell sites, base stations,access points, and/or any terminus of radio frequency communication.Although current network architectures may consider a distinctionbetween mobile/user devices and access points/cell sites, the exampleembodiments described hereafter may generally be applicable toarchitectures where that distinction is not so clear, such as ad hocand/or mesh network architectures, for example.

The algorithm determines a set of filter weights (and length) to reduce(cancel) the IQ imbalanced images. The algorithm uses a frequency-domainleast-squares fit method. Tone signals are sampled and input to ananalog circuit of a demodulator or modulator and, thus, the outputtedtone signals are imbalanced. Outputted tone samples for each tonesignal, at different frequencies, are captured. The algorithm determinesthe DFTs (Discrete Fourier Transform) of each sampled tone signal. Thefilter weights are adjusted to reproduce the tone and to eliminate theimage (target value of 0). The filter weights are determined based on aleast-squares fit on the DFT values (frequency-domain values) of theimbalanced sampled tones.

FIGS. 1A-1C illustrate a base station according to at least one exampleembodiment. FIG. 1A illustrates a base station 10 including a receiver20, a transmitter 30, a tone generator 40 and a microprocessor 50. Itshould be understood that the receiver 20 and the transmitter 30 may bea transceiver. However, for the sake of clarity they are illustrated asseparate elements.

As should be understood, the base station 10 is configured tocommunicate with a user equipment (UE) 60. The receiver 20 is configuredto receive signals and the transmitter is configured to transmitsignals. The tone generator 40 is configured to generate tones ofvarious frequencies as training signals for the receiver 20 and thetransmitter 30. The microprocessor 50 is a controller for the basestation 50. It should be understood that the base station 10 may includeadditional features and should not be limited to those shown in FIG. 1A.

FIG. 1B illustrates an example embodiment of the transmitter shown inFIG. 1A.

As shown in FIG. 1B, the transmitter includes a direct-path 70 a and animage-path 70 b. The paths 70 a, 70 b includes digital-to-analogconverters (DACs) 72 a, 72 b and low pass filters (LPFs) 74 a, 74 b anda modulator 78. An IQ pre-compensation system 76 pre-compensates aninput signal for IQ imbalance. At least one example embodiment of IQcompensation is described below with reference to FIGS. 3-4B. Thus, forthe sake of brevity, the IQ pre-compensation system 76 will not bedescribed in greater detail. However, it should be understood that theat least one example embodiment of FIGS. 3-4B may be user to determinethe pre-compensation systems 76 a, 76 b.

An input signal to be transmitted includes in-phase (real) signal and animage signal. As shown, both the real signal and image signal aresubject to the IQ pre-compensation system 76 and then pre-transmissionprocessing through the DACs 72 a, 72 b, LPFs 74 a, 74 b and themodulator 78. A signal output from the modulator 78 on the direct-path70 a is added to a signal output from the modulator 78 on the image-path70 b at an adder 80. The added signal output is input to a band passfilter (BPF) 82. The BPF 82 filters the added signal output.

The tone generator 40 is configured to generate tones of variousfrequencies as training signals for the transmitter 30.

FIG. 1C illustrates an example embodiment of the receiver shown in FIG.1A. As shown, the receiver is configured to receive a radio frequency(RF) signal. The tone generator 40 is configured to generate tones ofvarious frequencies as training signals for the receiver 20.

The receiver 20 includes a direct-path 83 a and an image-path 83 b. Eachpath 83 a, 83 b includes a LPF 86 a, 86 b, an operational amplifier 88a, 88 b and an analog-to-digital converter (ADC) 90 a, 90 b. A realsignal of the received signal is processed on the direct-path 83 a andan image signal of the received signal is processed on the image-path 83b. The ADC 90 a outputs in-phase samples of the received signal to an IQcompensator 92 and the ADC 90 b outputs image samples (quadraturesamples) to the IQ compensator 92.

At least one example embodiment of an IQ compensator is described withregards to FIGS. 3-4B. Thus, for the sake of brevity the IQ compensator92 will not be described in greater detail.

There are three equivalent channel models to describe IQ imbalance fromx(n) to z(n). The three models are a real IQ channel model, a complex IQchannel model and a direct/image channel model.

FIGS. 2A-2D illustrate examples where a signal received at a receiver isIQ Unbalanced. However, it should be understood the channel models shownin FIGS. 2A-2D may be applicable where a signal is IQ pre-compensatedbefore transmission.

Real IQ Channel Model

FIG. 2A illustrates a real IQ channel model according to an exampleembodiment. As shown, the sampled input signal x(n) is input to thechannel. The channel includes an I channel 210 a, which carries a realsignal of the sampled input signal x(n), and a Q channel 210 b, whichcarries an imaginary signal of the sampled input signal x(n). Thein-phase signal is input to real filters 212 a and 212 c and theimaginary signal is input to the real filters 212 b and 212D. Thefilters 212 a-212 d have weights h₁₁(m), h₁₂(m), h₂₁(m) and h₂₂(m),respectively, where m is 0, . . . , M−1, where M is the filter length.As should be understood, filter length is a number of impulse responsesamples in a filter.

The real filter 212 a may be referred to as the I channel through realfilter, the real filter 212 b may be referred to as the Q to I channelcross real filter, the real filter 212 c may be referred to as the I toQ channel cross real filter and the real filter 212 d may be referred toas the Q channel through real filter.

As shown, outputs from the real filters 212 a and 212 b are summed by anadder 215 onto the I channel 210 a. Outputs from the 212 c and 212 d aresummed by an adder 220 onto the Q channel 210 b.

Outputs from adders are offset by direct current (DC) offsets d_(in) andd_(q) at adders 222 and 224.

An output from the adder 224 is multiplied by j (=√{square root over(−1)}) and then added to an output from the adder 222 by an adder 230.The adder 230 outputs sampled imbalanced training signals y(n).

Complex IQ Channel Model

FIG. 2B illustrates a complex IQ channel model according to an exampleembodiment. As shown, the sampled input signal x(n) is input to thechannel. The channel includes an I channel 240 a, which carries thein-phase signal of the sampled input signal x(n), and a Q channel 240 b,which carries the quadrature-phase signal of the sampled input signalx(n). The in-phase signal is input to a complex filter 245 a and thequadrature-phase signal is input to a complex filter 245 b. The complexfilters 245A, 245B have weights h_(ip)(m) and h_(qp)(m), respectively,where

$\begin{matrix}\left\{ {{\begin{matrix}{{h_{ip}(m)} = {{h_{11}(m)} + {j\; {h_{21}(m)}}}} & {{m = 0},1,\ldots \mspace{14mu},\left( {M - 1} \right)} \\{{h_{qp}(m)} = {{h_{12}(m)} + {j\; {h_{22}(m)}}}} & \;\end{matrix}\begin{bmatrix}{h_{11}(m)} & {h_{12}(m)} \\{h_{21}(m)} & {h_{22}(m)}\end{bmatrix}} = \begin{bmatrix}{{Re}\left\{ {h_{ip}(m)} \right\}} & {{Re}\left\{ {h_{qp}(m)} \right\}} \\{{Im}\left\{ {h_{ip}(m)} \right\}} & {{Im}\left\{ {h_{qp}(m)} \right\}}\end{bmatrix}} \right. & (1)\end{matrix}$

An adder 250 adds the outputs from the complex filters 245 a and 245B.An adder 255 adds an output of the adder 255 to a complex DC offset,which is complex (d_(ip),d_(qp)).

The adder 255 outputs the sampled imbalanced training signals y(n).

Direct/Image Channel Model

FIG. 2C illustrates a direct/image channel model. As shown, the sampledinput signal x(n) is input to the channel. The channel includes a directpath 260 a, which carries a direct-path signal x_(d)(n) of the sampledinput signal x(n), and an image path 260 b, which carries the image-pathsignal x_(m)(n) of the sampled input signal x(n). The image-path signalx_(m)(n) is the complex conjugate of the direct-path signal x_(d)(n).The direct-path signal x_(d)(n) is input to an imbalance filter 265 aand the image-path signal x_(m)(n) is input to an imbalance filter 265b. The complex filters have weights h_(d)(m) and h_(m)(m), respectively,where

$\begin{matrix}\left\{ {{\begin{matrix}{{h_{d}(m)} = {{\left( {{h_{ip}(m)} - {j\; {h_{qp}(m)}}} \right)/2} = {\left( {\left( {{h_{11}(m)} + {h_{22}(m)}} \right) + {j\left( {{h_{21}(m)} - {h_{12}(m)}} \right)}} \right)/2}}} \\{{h_{m}(m)} = {{\left( {{h_{ip}(m)} + {j\; {h_{qp}(m)}}} \right)/2} = {\left( {\left( {{h_{11}(m)} - {h_{22}(m)}} \right) + {j\left( {{h_{21}(m)} + {h_{12}(m)}} \right)}} \right)/2}}}\end{matrix}\begin{bmatrix}{h_{11}(m)} & {h_{12}(m)} \\{h_{21}(m)} & {h_{22}(m)}\end{bmatrix}} = \begin{bmatrix}{{Re}\left\{ {{h_{m}(m)} + {h_{d}(m)}} \right\}} & {{Im}\left\{ {{h_{m}(m)} - {h_{d}(m)}} \right\}} \\{{Im}\left\{ {{h_{d}(m)} + {h_{m}(m)}} \right\}} & {{- {Re}}\left\{ {{h_{m}(m)} - {h_{d}(m)}} \right\}}\end{bmatrix}} \right. & (2)\end{matrix}$

An adder 270 adds the outputs from the imbalance filters 265 a and 265.An adder 275 adds an output of the adder 275 to a complex DC offset.

The adder 275 outputs the sampled imbalanced training signals y(n).

From FIGS. 2A-C, each of the filters may be converted to another filteras shown in FIG. 2D.

IQ Imbalance Compensation

FIG. 3 illustrates an IQ imbalance and compensation model according toat least one example embodiment. As shown in FIG. 3, a compensationmodel 310 is configured to receive the sampled imbalanced trainingsignals y(n) output from a channel estimation model 320. The channelestimation model 320 is the direct/image channel model shown in FIG. 2C.It should be understood that the compensation model 310 may beimplemented as the IQ compensator 92, shown in FIG. 1C, and may bemodified to function as the IQ pre-compensator 76, shown in FIG. 1B.

The direct-path signal is the imbalanced training signal y(n) and theimage-path signal is the conjugate of the imbalanced training signal, orConj{y(n)}. While at least one example embodiment is described belowwith reference to direct-path and image-path signals, the algorithm isalso applicable to real IQ channel model and complex IQ channel model

As shown, the compensation model 310 includes a direct path 325 a and animage path 325 b. A compensation filter w_(d) 330 a is on the directpath 325 a and receives the imbalanced training signal (direct-pathsignal) y(n). A compensation filter w_(m) 330 b is on the image path 325b and receives the complex conjugate of the imbalanced training signaly(n). The compensation filters 330 a and 330 b have weights w_(d)(m) andw_(m)(m), respectively.

The compensation filter weights w_(d)(m) and w_(m)(m) are determined bythe microprocessor to compensate for the IQ imbalance of the imbalancedsignal y(n), as will be described below.

As shown, the input sampled signal x(n) includes a direct-path signalx_(d)(n) and an image-path signal x_(m)(n), which is the complexconjugate of x(n). The Discrete Fourier Transform (DFT) for thedirect-path imbalance filter 265 a is determined by

H _(d)(k)=Σ_(n=0) ^(N−1) h _(d)(n)·e ^(−j2πkn/N)   (3)

and the image-path imbalance filter 265 b DFT is determined by

H _(m)(k)=Σ_(n=0) ^(N−1) h _(m)(n)·e ^(−j2πkn/N)   (4)

where N is a DFT bin size, the filter length M is less than the DFT binsize N and k is from zero to N−1. The input sampled signal x(n) isanalyzed to determine the DFTs of the direct-path imbalance filter 265 aand the image-path imbalance filter 265 b.

The continuous wave tone training signal x(t) is a complex exponentialwith a known frequency, but unknown amplitude and phase. The continuouswave tone training signal x(t) is not observable. For analysis purposes,the sampled training signal x(n), although still not observable, isconsidered where the sampling rate is, for example, at 61.44 MHz rate.In an example embodiment described below, the input sampled signal x(n)is a sampled tone signal generated by the tone generator 40.

It should be understood that a wideband signal may be used instead ofthe tone signal to train the filter weights. The frequency band of thewideband signal is preset so that it does not overlap with an imagesignal frequency band. Multiple captures of the wideband signals ofdifferent frequency bands are gathered to cover the whole frequency bandof interest. In the base station 10, the tone generator 40 is programmedto generate the continuous wave tone training signal x(t) looped back tothe receiver 20. Therefore, the base station 10 considers tone-trainedIQ imbalance compensation.

For example, the tone generator 40 generates a tone with a frequencyoffset df_(i), where i is a tone index. For example, table 1 illustratestone index i associated with a frequency offset df_(i), a localoscillator frequency of the tone generator and local oscillatorfrequency of the receiver.

TABLE 1 Tone Frequency Generator Rx Tone Offset df_(i) LO LO Index, i(kHz) (kHz) (kHz) 0 −9,900 2,525,100 2,535,000 1 −7,500 2,527,5002,535,000 2 −5,100 2,529,900 2,535,000 3 −2,700 2,532,300 2,535,000 4−300 2,534,700 2,535,000 5 300 2,535,300 2,535,000 6 2,700 2,537,7002,535,000 7 5,100 2,540,100 2,535,000 8 7,500 2,542,500 2,535,000 99,900 2,544,900 2,535,000

The Rx LO in column 5 of Table 1 is a frequency of the local oscillatorin a demodulator of a receiver. For example, in FIG. 1C, the Rx LO isthe frequency of the local oscillator in the demodulator 84. Thefrequency offset is the frequency of the continuous wave tone trainingsignal x(t).

The direct-path input sampled signal x_(d)(n) is

x _(d)(n)=x(n)=a_(i) ·e ^(j2πk) ^(i) ^(n/N)   (5)

where a_(i) is the tone initial phase and amplitude for the index i.

The tone initial phase and amplitude a_(i) is a complex number andunknown. Index k_(i) is the DFT bin index associated with tone index iwhere the angle of the sine wave at sample time n is

2Πk_(i)n/N   (6)

The DFT for the direct-path input sampled signal is:

$\begin{matrix}\begin{matrix}{{X_{d}(k)} = {X(k)}} \\{= {\sum\limits_{n = 0}^{N - 1}{{x(n)} \cdot ^{{- {j2\pi}}\; k\; {n/N}}}}} \\{= {a_{i}{\sum\limits_{n = 0}^{N - 1}^{{- {j2\pi}}\; {({k - k_{i}})}{n/N}}}}} \\{= \left\{ \begin{matrix}{a_{i}N} & {k = k_{i}} \\{\approx 0} & {k \neq k_{i}}\end{matrix} \right.}\end{matrix} & (7)\end{matrix}$

Similarly, the image-path input sampled signal x_(m)(n) and DFT are

$\begin{matrix}{{x_{m}(n)} = {{x^{*}(n)} = {a_{i}^{*\;} \cdot ^{{- j}\; 2\pi \; k_{i}{n/N}}}}} & (8) \\\begin{matrix}{{X_{m}(k)} = {\sum\limits_{n = 0}^{N - 1}{a_{i}^{*} \cdot ^{{- {j2\pi}}\; k_{i}{n/N}} \cdot ^{j\; 2\pi \; {{kn}/N}}}}} \\{= {a_{i}^{*}{\sum\limits_{n = 0}^{N - 1}^{{- {j2\pi}}\; {({k + k_{i}})}{n/N}}}}} \\{= \left( {X\left( {- k} \right)} \right)^{*}} \\{= \left\{ \begin{matrix}{a_{i}^{*}N} & {k = {- k_{i}}} \\{\approx 0} & {k \neq {- k_{i}}}\end{matrix} \right.}\end{matrix} & (9)\end{matrix}$

The imbalanced input signal y(n) is observable and, thus, may be used bythe processor 50 to estimate the imbalance and compensation filters.Based on equations (5)-(6) and (8)-(9), the DFT of the imbalanced inputsignal y(n) may be related to the input signals and filters. The timeseries and DFT values of the imbalanced input signal are, where

denotes convolution

$\begin{matrix}{{y(n)} = {{{h_{d}(n)} \otimes {x_{d}(n)}} + {{h_{m}(n)} \otimes {x_{m}(n)}}}} & (10) \\\begin{matrix}{{Y(k)} = {{{H_{d}(k)}{X_{d}(k)}} + {{H_{m}(k)}{X_{m}(k)}}}} \\{= \left\{ \begin{matrix}{a_{i}{{NH}_{d}\left( k_{i} \right)}} & {k = k_{i}} \\{a_{i}^{*}{{NH}_{m}\left( k_{i} \right)}} & {k = {- k_{i}}} \\{\approx 0} & {else}\end{matrix} \right.}\end{matrix} & (11)\end{matrix}$

Equation (11) shows that the DFT Y(k) is a sum of impulses at positiveand negative frequencies of the input training tone index i where theDFT bin index k is at the positive index k_(i) and at the negative index−k_(i).

As stated above, the initial phase and amplitude a_(i) of the sampledinput training tone signal x(n) is initially unknown. However, theinitial phase and amplitude a_(i) may be determined from the DFT of thein-phase signal or the quadrature-phase signal. For the sake of brevity,an algorithm using the DFT of the in-phase signal is described. Thein-phase signal r(n) in time series and DFT are determined as follows:

x(n)==a _(i) e ^(j2πk) ^(i) ^(n/N) =|a _(i) |e ^((jθ+j2πnk) ^(i) ^(n/N))  (12)

where

a _(i) =|a _(i) |e ^(jθ)  (13)

thus

$\begin{matrix}{{r(n)} = {{{Re}\left( {x(n)} \right)} = {\frac{a_{i}}{2}\left( {^{({{j\; \theta} + {{j2}\; \pi \; n\; k_{i}{n/N}}})} + ^{- {({{j\; \theta} + {j\; 2\pi \; n\; k_{i}{n/N}}})}}} \right)}}} & (14)\end{matrix}$

The DFT of the in-phase signal is determined by

$\begin{matrix}\begin{matrix}{{R(k)} \equiv {\sum\limits_{n = 0}^{N - 1}{{r(n)} \cdot ^{{- j}\; 2\; \pi \; k\; {n/N}}}}} \\{= {\sum\limits_{n = 0}^{N - 1}{\frac{a_{i}}{2}{\left( {^{({{j\; \theta} + {{j2}\; \pi \; n\; k_{i}{n/N}}})} + ^{- {({{j\; \theta} + {j\; 2\pi \; n\; k_{i}{n/N}}})}}} \right) \cdot ^{{- j}\; 2\; \pi \; k\; {n/N}}}}}} \\{= {{\frac{a_{i}}{2}^{j\; \theta}{\sum\limits_{n = 0}^{N - 1}^{{- j}\; 2\; {\pi {({k - k_{i}})}}\; {n/N}}}} + {\frac{a_{i}}{2}^{{- j}\; \theta}{\sum\limits_{n = 0}^{N - 1}^{{- j}\; 2\; {\pi {({k + k_{i}})}}\; {n/N}}}}}} \\{= \left\{ \begin{matrix}{a_{i}{N/2}} & {k = k_{i}} \\{a_{i}^{*}{N/2}} & {k = {- k_{i}}} \\{\approx 0} & {else}\end{matrix} \right.}\end{matrix} & (15)\end{matrix}$

As shown in equation (15), the initial phase and amplitude a_(i) isdetermined from the in-phase DFT value R(k_(i)) using

a _(i) N=2R(k _(i))=2(R(−k _(i)))*   (16)

Therefore, the initial phase and amplitude a_(i) can be determined basedon the in-phase DFT at the positive and negative frequencies associatedthe DFT bin indexes k_(i) and −k_(i) in equation (16).

By relating the DFT of the in-phase signal R(k) to the DFT of theobservation signal Y(k), the DFTs of both the direct-path imbalancefilter 265 a and the image-path imbalance filter 265 b can be expressedin terms of known quantities. The DFT of the in-phase signal y_(i)(n) atthe positive bin index k_(i) is

Y(k _(i))=a _(i) NH _(d)(k _(i))=2R(k _(i))H _(d)(k _(i))   (17)

The DFT of the image signal y_(q)(n) at the negative bin index −k_(i) is

Y(−k _(i))=a* _(i) NH _(m)(−k _(i))=2(R(−k _(i)))*H _(m)(−k _(i))   (18)

Expanding the filter DFT terms in equation (17), the direct-pathimbalance filter weights h_(d)(m) 265 a satisfies the following linearequations (i=0, 1, . . . , I−1)

$\begin{matrix}{\frac{Y\left( k_{i} \right)}{2{R\left( k_{i} \right)}} = {{H_{d}\left( k_{i} \right)} = {\sum\limits_{m = 0}^{M - 1}{{h_{d}(m)}^{{- j}\; 2\; \pi \; k_{i}{m/N}}}}}} & (19)\end{matrix}$

Expanding equation (18), the image-path imbalance filter weightsh_(m)(m) 265 b satisfies the following linear equations (i=0, 1, . . . ,I−1)

$\begin{matrix}{\frac{Y\left( {- k_{i}} \right)}{2\left( {R\left( {- k_{i}} \right)} \right)^{*}} = {{H_{m}\left( {- k_{i}} \right)} = {\sum\limits_{m = 0}^{M - 1}{{h_{m}(m)}^{j\; 2\; \pi \; k_{i}{m/N}}}}}} & (20)\end{matrix}$

The DFT values Y(±k_(i)) and R(±k_(i)) are determined from the observeddata of the imbalanced signal y(n). The variables in the above equations(19) and (20) are the channel estimate filter weights h_(d)(m) andh_(m)(m).

Equation (19) is put into matrix format to determine a least-squaressolution for an estimate of the direct-path imbalance filter 265 a asfollows:

$\begin{matrix}{\mspace{79mu} {u = ^{j\; 2{\pi/N}}}} & (21) \\{\mspace{79mu} {{A_{d}h_{d}} = b_{d}}} & (22) \\{\begin{bmatrix}1 & u^{- k_{0}} & u^{{- 2}k_{0}} & \ldots & u^{{- {({M - 1})}}k_{0}} \\1 & u^{- k_{1}} & u^{{- 2}k_{1}} & \ldots & u^{{- {({M - 1})}}k_{1}} \\1 & u^{- k_{2}} & u^{{- 2}k_{2}} & \ldots & u^{{- {({M - 1})}}k_{2}} \\\vdots & \vdots & \vdots & \ddots & \vdots \\1 & u^{- k_{I - 1}} & u^{{- 2}k_{I - 1}} & \ldots & u^{{- {({M - 1})}}k_{I - 1}}\end{bmatrix}{\quad{\begin{bmatrix}{h_{d}(0)} \\{h_{d}(1)} \\{h_{d}(2)} \\\vdots \\{h_{d}\left( {M - 1} \right)}\end{bmatrix} = \begin{bmatrix}{{{Y\left( k_{0} \right)}/2}{R\left( k_{0} \right)}} \\{{{Y\left( k_{1} \right)}/2}{R\left( k_{1} \right)}} \\{{{Y\left( k_{2} \right)}/2}{R\left( k_{2} \right)}} \\\vdots \\{{{Y\left( k_{I - 1} \right)}/2}{R\left( k_{I - 1} \right)}}\end{bmatrix}}}} & (23)\end{matrix}$

The least-squares solution for the direct-path imbalance filter 265 a is

ĥ _(d)=(A _(d) ^(H) A _(d))⁻¹ A _(d) ^(H) b _(d)   (23)

Equation (20) is put into matrix format to determine a least-squaressolution for an estimate of the image-path imbalance compensation filter265 b as follows:

$\begin{matrix}{\mspace{79mu} {{A_{m}h_{m}} = b_{m}}} & (25) \\{\begin{bmatrix}1 & u^{k_{0}} & u^{2k_{0}} & \ldots & u^{{({M - 1})}k_{0}} \\1 & u^{k_{1}} & u^{2k_{1}} & \ldots & u^{{({M - 1})}k_{1}} \\1 & u^{k_{2}} & u^{2k_{2}} & \ldots & u^{{({M - 1})}k_{2}} \\\vdots & \vdots & \vdots & \ddots & \vdots \\1 & u^{k_{I - 1}} & u^{2k_{I - 1}} & \ldots & u^{{({M - 1})}k_{I - 1}}\end{bmatrix}{\quad{\begin{bmatrix}{h_{m}(0)} \\{h_{m}(1)} \\{h_{m}(2)} \\\vdots \\{h_{m}\left( {M - 1} \right)}\end{bmatrix} = \begin{bmatrix}{{{Y\left( {- k_{0}} \right)}/2}{R\left( k_{0} \right)}^{*}} \\{{{Y\left( {- k_{1}} \right)}/2}{R\left( k_{1} \right)}^{*}} \\{{{Y\left( {- k_{2}} \right)}/2}{R\left( k_{2} \right)}^{*}} \\\vdots \\{{{Y\left( {- k_{I - 1}} \right)}/2}{R\left( k_{I - 1} \right)}^{*}}\end{bmatrix}}}} & (26)\end{matrix}$

where

I≧M   (27)

The least-squares solution for the image-path imbalance filter 265 b is

ĥ _(m)=(A _(m) ^(H) A _(m))⁻¹ A _(m) ^(H) b _(m)   (26)

The IQ imbalance compensation filters 330 a and 330 b may be determinedwithout computing the IQ imbalance channel estimate, which is describedabove.

As shown in FIG. 3, the configuration of the IQ imbalance compensationhas two filters: the direct-path filter w_(d)(m) and image-path filterw_(m)(m). N-point DFTs of the direct-path compensation filter 330 a andthe image-path compensation filter 330 b are given by

W _(d)(k)=Σ_(n=0) ^(N−1) w _(d)(n)·e ^(−j2πkn/N)   (29)

W _(m)(k)=Σ_(n=0) ^(N−1) w _(n)(n)·e ^(−j2πkn/N)   (30)

k=0, . . . , N−1The length M of the compensation filters 330 a and 330 b is less thanthe DFT bin size N.

The IQ compensated output signal z(n) is the sum of the filtereddirect-path signal (output of the direct-path compensation filter 330 a)and image-path signals (output of the image-path compensation filter 330b). A DFT of the output signal z(n) is expressed as follows:

z(n)=y _(d)(n)

w _(d)(n)+y _(m)(n)

w _(m)(n)   (31)

Z(k)=Y _(d)(k)W _(d)(k)+Y _(m)(k)W _(m)(k)   (32)

The DFT value of Z(k) may be set to the tone DFT value at the tone indexand to set to zero at the image signal index, as shown in equations (33)and (34) below:

Z(k _(i))=W _(d)(k _(i))Y(k _(i))+W _(m)(k _(i))(Y(−k _(i)))*=a _(i)N=2R(k _(i))e ^(−j2πk) ^(i) ^(d/N)   (33)

at k=k_(i),

Z(−k _(i))=W _(d)(−k _(i))Y(−k _(i))+W _(m)(−k _(i))(Y(k _(i)))*=0  (34)

at k=−k_(i).where delay d=M/2.

As described above, filter weights w_(d)(m) and w_(m)(m) are determinedsuch that the original tone signal is preserved and the image signal isreduced. Thus, the right hand side of equation (33) is set to the toneinitial phase and amplitude and the right hand side of equation (34) isset to equal zero.

In equation (33), the initial phase and amplitude a_(i) is determinedfrom the in-phase DFT value R(k_(i)) using

a _(i) N=2R(k _(i))=2(R(−k _(i)))*   (35)

Therefore, equations (33) and (34) are expanded as a system of equationsin terms of the compensation filter weights w_(d) and w_(m).

Y(k _(i))Σ_(n=0) ^(M−1) w _(d)(m)e ^(−j2πk) ^(i) ^(m/N)+(Y(−k_(i)))*Σ_(m=0) ^(M−1) w _(m)(m)e ^(−j2πk) ^(i) ^(m/N)=2R(k _(i))e^(−j2πk) ^(i) ^(d/N)   (36)

Y(−k _(i))Σ_(m=0) ^(M−1) w _(d)(m)e ^(j2πk) ^(i) ^(m/N)+(Y(k_(i)))*Σ_(m=0) ^(M−1) w _(m)(m)e ^(j2πk) ^(i) ^(m/N)=0   (37)

i=0,1, . . . , (I−1), I≧Mwhere delay d=integer(M/2).

Equations (36) and (37) may be written in matrix format, by definingfour matrices

$\mspace{20mu} {A = \begin{bmatrix}{Y\left( k_{0} \right)} & {{Y\left( k_{0} \right)}u^{- k_{0}}} & \ldots & {{Y\left( k_{0} \right)}u^{{- {({M - 1})}}k_{0}}} \\{Y\left( k_{1} \right)} & {{Y\left( k_{1} \right)}u^{- k_{1}}} & \ldots & {{Y\left( k_{1} \right)}u^{{- {({M - 1})}}k_{1}}} \\\vdots & \vdots & \ddots & \vdots \\{Y\left( k_{I - 1} \right)} & {{Y\left( k_{I - 1} \right)}u^{- k_{I - 1}}} & \ldots & {{Y\left( k_{I - 1} \right)}u^{{- {({M - 1})}}k_{I - 1}}}\end{bmatrix}}$ $B = \begin{bmatrix}\left( {Y\left( {- k_{0}} \right)} \right)^{*} & {\left( {Y\left( {- k_{0}} \right)} \right)^{*}u^{- k_{0}}} & \ldots & {\left( {Y\left( {- k_{0}} \right)} \right)^{*}u^{{- {({M - 1})}}k_{0}}} \\\left( {Y\left( {- k_{1}} \right)} \right)^{*} & {\left( {Y\left( {- k_{1}} \right)} \right)^{*}u^{- k_{1}}} & \ldots & {\left( {Y\left( {- k_{1}} \right)} \right)^{*}u^{{- {({M - 1})}}k_{1}}} \\\vdots & \vdots & \ddots & \vdots \\\left( {Y\left( {- k_{I - 1}} \right)} \right)^{*} & {\left( {Y\left( {- k_{I - 1}} \right)} \right)^{*}u^{- k_{I - 1}}} & \ldots & {\left( {Y\left( {- k_{I - 1}} \right)} \right)^{*}u^{{- {({M - 1})}}k_{I - 1}}}\end{bmatrix}$ $\mspace{20mu} {C = \begin{bmatrix}{Y\left( {- k_{0}} \right)} & {{Y\left( {- k_{0}} \right)}u^{k_{0}}} & \ldots & {{Y\left( {- k_{0}} \right)}u^{{({M - 1})}k_{0}}} \\{Y\left( {- k_{1}} \right)} & {{Y\left( {- k_{1}} \right)}u^{k_{1}}} & \ldots & {{Y\left( {- k_{1}} \right)}u^{{({M - 1})}k_{1}}} \\\vdots & \vdots & \ddots & \vdots \\{Y\left( {- k_{I - 1}} \right)} & {{Y\left( {- k_{I - 1}} \right)}u^{k_{I - 1}}} & \ldots & {{Y\left( k_{I - 1} \right)}u^{{({M - 1})}k_{I - 1}}}\end{bmatrix}}$ $\mspace{20mu} {D = \begin{bmatrix}\left( {Y\left( k_{0} \right)} \right)^{*} & {\left( {Y\left( k_{0} \right)} \right)^{*}u^{k_{0}}} & \ldots & {\left( {Y\left( k_{0} \right)} \right)^{*}u^{{({M - 1})}k_{0}}} \\\left( {Y\left( k_{1} \right)} \right)^{*} & {\left( {Y\left( k_{1} \right)} \right)^{*}u^{k_{1}}} & \ldots & {\left( {Y\left( k_{1} \right)} \right)^{*}u^{{({M - 1})}k_{1}}} \\\vdots & \vdots & \ddots & \vdots \\\left( {Y\left( k_{I - 1} \right)} \right)^{*} & {\left( {Y\left( k_{I - 1} \right)} \right)^{*}u^{k_{I - 1}}} & \ldots & {\left( {Y\left( k_{I - 1} \right)} \right)^{*}u^{{({M - 1})}k_{I - 1}}}\end{bmatrix}}$

Aggregating matrices A, B, C and D forms a matrix on the left hand side

$U \equiv \begin{bmatrix}A & B \\C & D\end{bmatrix}$

The right hand side vector is

$b = \begin{bmatrix}{2{R\left( k_{0} \right)}u^{{- k_{0}}d}} \\{2{R\left( k_{1} \right)}u^{{- k_{1}}d}} \\\vdots \\{2{R\left( k_{I - 1} \right)}u^{{- k_{I - 1}}d}}\end{bmatrix}$

I≧M where u=e^(j2π/N) and the delay d=integer(M/2).

The variables are the compensation filter weights w_(d) and w_(m)

${w_{d} = \begin{bmatrix}{w_{d}(0)} \\{w_{d}(1)} \\\vdots \\{w_{d}\left( {M - 1} \right)}\end{bmatrix}},{w_{m} = \begin{bmatrix}{w_{m}(0)} \\{w_{m}(1)} \\\vdots \\{w_{m}\left( {M - 1} \right)}\end{bmatrix}}$

Consequently, the system of equations for the compensation filters 330 aand 330 b is

$\begin{matrix}{{\begin{bmatrix}A & B \\C & D\end{bmatrix}\begin{bmatrix}w_{d} \\w_{m}\end{bmatrix}} = {{\begin{bmatrix}b \\0\end{bmatrix}\mspace{14mu} {or}\mspace{14mu} {U\begin{bmatrix}w_{d} \\w_{m}\end{bmatrix}}} = \begin{bmatrix}b \\0\end{bmatrix}}} & (38)\end{matrix}$

The least-squares solution for the IQ imbalance compensation filterweights w_(d) and w_(m) is

$\begin{matrix}{\begin{bmatrix}w_{d} \\w_{m}\end{bmatrix} = {\left( {U^{H}U} \right)^{- 1}{U^{H}\begin{bmatrix}b \\0\end{bmatrix}}}} & (39)\end{matrix}$

Alternatively, the matrices A, B, C and D may be determined as follows

$\begin{matrix}{{E = \begin{bmatrix}1 & u^{- k_{0}} & \ldots & u^{{- {({M - 1})}}k_{0}} \\1 & u^{- k_{1}} & \ldots & u^{{- {({M - 1})}}k_{1}} \\\vdots & \vdots & \ddots & \vdots \\1 & u^{- k_{I - 1}} & \ldots & u^{{- {({M - 1})}}k_{I - 1}}\end{bmatrix}}{Y_{d} = {{\begin{bmatrix}{Y\left( k_{0} \right)} \\{Y\left( k_{1} \right)} \\\vdots \\{Y\left( k_{I - 1} \right)}\end{bmatrix}\mspace{20mu} Y_{m}} = \begin{bmatrix}{Y\left( {- k_{0}} \right)} \\{Y\left( {- k_{1}} \right)} \\\vdots \\{Y\left( {- k_{I - 1}} \right)}\end{bmatrix}}}{A = {{{diag}\left( Y_{d} \right)} \cdot E}}} & (40) \\{B = {{{diag}\left( \left( Y_{m} \right)^{*} \right)} \cdot E}} & (41) \\{C = {{{diag}\left( Y_{m} \right)} \cdot E^{*}}} & (42) \\{D = {{{diag}\left( \left( Y_{d} \right)^{*} \right)} \cdot E^{*}}} & (43)\end{matrix}$

As described above, the filter compensation weights are determined arethen may be used to filter received signals and reduce the IQ imbalance.For example, as shown in FIG. 1C, the IQ compensation system 92 includesa direct-path IQ compensation filter and an image-path IQ compensationfilter. The weights of the direct-path IQ compensation filter and animage-path IQ compensation filter are determined using the algorithmdescribed above with reference to FIG. 3. Consequently, the receiver 20may receive signals transmitted from a remote device (e.g., the UE 60)and the IQ compensation system may reduce the IQ imbalance from thereceived signal by filtering the received signal after it is processedby the analog circuits (the modulator 84, low pass filters, 86 a, 86 b,operational amplifiers 88 a, 88 b and ADCs 90 a, 90 b).

FIG. 4A illustrates the compensation model 310 in more detail. A realsignal is input to real filters 335 a and 335 c and the imaginary signalis input to the real filters 335 b and 335 b. The filters 335 a-335 dhave weights w₁₁(m), w₁₂(m), w₂₁(m) and w₂₂(m), respectively where m is0, . . . , M−1, where M is the filter length. The weights w₁₁(m),w₁₂(m), w₂₁(m) and w₂₂(m) may be converted to the weights w_(d) andw_(m) using the same conversion methods described with reference toFIGS. 2A-2D. Thus, for the same of brevity they will not be described.

As shown, outputs from the real filters 335 a and 335 b are summed by anadder 340 a onto the direct-path 325 a. Outputs from the 335 c and 335 dare summed by an adder 340 b onto the image-path 340 b.

An output from the adder 340 b is multiplied by j (=√{square root over(−1)}) by a multiplier 345 and then added to the output from the adder340 a by an adder 350. The adder 350 outputs a compensated output signalz(n).

To reduce implementation resources, the four real filters 335 a-335 dmay be reduced to two real filters by rotating and adjusting a gain.Matrix rotation is used in matrix operations. It is described inhttp://mathworld.wolfram.com/RotationMatrix.html using a matrix thatrotates the coordinate system through an angle. For example,

${{w(z)} = \begin{bmatrix}{w_{11}(z)} & {w_{12}(z)} \\{w_{21}(z)} & {w_{22}(z)}\end{bmatrix}},{{u(z)} = \begin{bmatrix}{u_{I}(z)} \\{u_{Q}(z)}\end{bmatrix}},{{v(z)} = \begin{bmatrix}{v_{I}(z)} \\{v_{Q}(z)}\end{bmatrix}}$

where w(z) is the z-transforms of the weights of the filters 335 a-335d, u(z) is the z-transform of u(n), which is an input signal to thefilter matrix w. And, v(n) is the output of the filter w. Therefore,

$\begin{matrix}{{v(z)} = {{w(z)}{u(z)}\mspace{14mu} {or}}} & (44) \\{\begin{bmatrix}{v_{I}(z)} \\{v_{Q}(z)}\end{bmatrix} = {\begin{bmatrix}{w_{11}(z)} & {w_{12}(z)} \\{w_{21}(z)} & {w_{22}(z)}\end{bmatrix}\begin{bmatrix}{u_{I}(z)} \\{u_{Q}(z)}\end{bmatrix}}} & (45)\end{matrix}$

The elements of w(z) are rotated with an angle function θ(z), where

$\begin{matrix}{{\tan \left( {\theta (z)} \right)} = \frac{w_{12}(z)}{w_{22}(z)}} & (46)\end{matrix}$

thus,

$\begin{matrix}{{{c(z)} = {{\cos \left( {\theta (z)} \right)} = \frac{w_{22}(z)}{\sqrt{{w_{22}(z)}^{2} + {w_{22}(z)}^{2}}}}}{{s(z)} = {{\sin \left( {\theta (z)} \right)} = \frac{w_{12}(z)}{\sqrt{{w_{22}(z)}^{2} + {w_{22}(z)}^{2}}}}}} & (47)\end{matrix}$

and, therefore,

$\begin{matrix}{{\begin{bmatrix}{c(z)} & {- {s(z)}} \\{s(z)} & {c(z)}\end{bmatrix}\begin{bmatrix}{w_{11}(z)} & {w_{12}(z)} \\{w_{21}(z)} & {w_{22}(z)}\end{bmatrix}} = \begin{bmatrix}{w_{11}^{\prime}(z)} & 0 \\{w_{21}^{\prime}(z)} & {w_{22}^{\prime}(z)}\end{bmatrix}} & (48)\end{matrix}$

since

c(z)w ₁₂(z)−s(z)w ₂₂(z)=(w ₂₂(z)w ₁₂(z)−w ₁₂(z)w ₂₂(z))/√{square rootover (w ₂₂(z)² +w ₂₂(z)²)}{square root over (w ₂₂(z)² +w ₂₂(z)²)}=0

The gain normalized by w′₁₁(z) with a delay d such that

$\begin{matrix}{{{\frac{z^{- d}}{w_{11}^{\prime}(z)}\begin{bmatrix}{c(z)} & {- {s(z)}} \\{s(z)} & {c(z)}\end{bmatrix}}\begin{bmatrix}{w_{11}(z)} & {w_{12}(z)} \\{w_{21}(z)} & {w_{22}(z)}\end{bmatrix}} = {{\frac{z^{- d}}{w_{11}^{\prime}(z)}\begin{bmatrix}{w_{11}^{\prime}(z)} & 0 \\{w_{21}^{\prime}(z)} & {w_{22}^{\prime}(z)}\end{bmatrix}} = \begin{bmatrix}z^{- d} & 0 \\{w_{21}^{''}(z)} & {w_{22}^{''}(z)}\end{bmatrix}}} & (50)\end{matrix}$

where

w′ ₁₁(z)=c(z)w ₁₁(z)−s(z)w ₂₁(z), w″ ₂₁(z)=w′ ₂₁(z)/w′ ₁₁(z), w″₂₂(z)=w′ ₂₂(z)/w′ ₁₁(z)   (51)

The delay is one half of the filter length M. Reducing equations (50)and (51) results in,

$\begin{matrix}{{y(z)} = {\frac{z^{- d}}{w_{11}^{\prime}(z)}\begin{bmatrix}{c(z)} & {- {s(z)}} \\{s(z)} & {c(z)}\end{bmatrix}}} & (52)\end{matrix}$

and

$\begin{matrix}{{v(z)} = {\begin{bmatrix}z^{- d} & 0 \\{w_{21}^{''}(z)} & {w_{22}^{''}(z)}\end{bmatrix}\begin{bmatrix}{u_{I}(z)} \\{u_{Q}(z)}\end{bmatrix}}} & (53)\end{matrix}$

Z^(−d) is the z-transform of the unit impulse of delay d.

As shown in equation (53), v(z) utilizes two filters. By rotating andadjusting the gain, the four real filters 335 a-335 d are reduced to tworeal filters 336 a-336 b, shown in FIG. 4B. Furthermore, the direct-path325 a is the in-phase signal of the imbalanced signal y(n) delayed by adelay 337 a.

The image-path 325 b includes the real filters 336 a and 336 b and adelay 337 b. The real filter 336 a receives crossover from the realsignal. The real filter 336 b and the delay 337 b receive the imaginarysignal. The real filters 336 a and 336 b compensate for the IQ imbalanceof the imbalanced signal y(n). The outputs of the real filters 336 a and336 b and the delay 336 b are summed by the adder 340 b and thenmultiplied by j by the multiplier 345. The output of the multiplier 345is added to the output of the delay 337 a by the adder 350. The adder350 outputs the compensated output signal z(n).

Using the models as explained above, filter weights may be determined bythe microprocessor in the base station using the IQ compensationalgorithm described above.

At least one example embodiment discloses an algorithm for determining aset of filter weights (and length) to reduce (cancel) the IQ imbalancedimages. The algorithm includes a frequency-domain least-squares fit.Tone signals are sampled and input to an analog circuit of a demodulatoror modulator and, thus, the outputted tone signals are imbalanced.Outputted tone samples for each tone signal are captured at differentfrequencies. The algorithm determines the DFTs (Discrete FourierTransform) of each sampled tone signal. The filter weights are adjustedto reproduce the tone and to eliminate the image (target value of 0).The filter weights are determined based on a least-squares fit on theDFT values (frequency-domain values) of the imbalanced sampled tones.

Moreover, at least one example embodiment discloses a least-squares fitin the frequency doming. At least one example embodiment discloses tworeal filters instead of 4 real filters.

Example embodiments being thus described, it will be obvious that thesame may be varied in many ways. Such variations are not to be regardedas a departure from the spirit and scope of example embodiments, and allsuch modifications as would be obvious to one skilled in the art areintended to be included within the scope of the claims.

1. A method of compensating for in-phase and quadrature (IQ) imbalancein a base station, the method comprising: generating, at the basestation, compensation filter weights based on a plurality of IQimbalanced training signals, the generating including, determining thecompensation filter weights based on the plurality of imbalancedtraining signals in a frequency domain; and filtering based on thecompensation filter weights.
 2. The method of claim 1, furthercomprising: generating input signals for an analog circuit of the basestation, the IQ imbalanced training signals being outputs of the analogcircuit and based on the input signals.
 3. The method of claim 2,wherein the generating input signals includes generating continuous wavetone signals as the input signals.
 4. The method of claim 2, wherein thegenerating input signals includes generating wideband signals as theinput signals.
 5. The method of claim 2, wherein the generating inputsignals includes, determining Discrete Fourier Transforms (DFT) of areal signal of the input signal.
 6. The method of claim 5, the filteringincludes, receiving a signal from a remote device, and filtering thesignal to reduce an IQ imbalance of the signal.
 7. The method of claim6, wherein the filtering includes, outputting a compensated outputsignal, the compensated output signal being based on a DFT of thereceived signal.
 8. The method of claim 7, wherein the compensatedoutput signal is two times the DFT of the real signal of the receivedsignal.
 9. The method of claim 1, wherein the determining thecompensation filter weights includes, determining DFTs of the IQimbalanced training signals.
 10. The method of claim 9, wherein thedetermining the compensation filter weights includes, determining thecompensation filter weights based on a least-squares fit of the DFTs ofthe IQ imbalanced training signals.
 11. The method of claim 9, whereineach IQ imbalanced training signal has a direct-path signal and an imagesignal, the determining the DFTs of the IQ imbalance training signalsincludes, determining the DFTs of the direct-path signals and image-pathsignals of the IQ imbalanced training signals.
 12. The method of claim11, further comprising: filtering at least one of the plurality of IQimbalanced signals using the compensation filter weights.
 13. The methodof claim 12, wherein the filtering includes, digitally filtering atleast one of the plurality of IQ imbalanced signals using thecompensation filter weights.
 14. A base station configured to generatecompensation filter weights based on a plurality of IQ imbalancedtraining signals, wherein the generating includes, determining thecompensation filter weights based on the plurality of imbalancedtraining signals in a frequency domain.
 15. The base station of claim14, wherein the base station is configured to generate compensationfilter weights using two filters.
 16. A user equipment (UE) configuredto receive a compensated signal from a base station, the signal beingcompensated based on an plurality of IQ imbalanced training signals andcompensation filter weights based on the plurality of imbalancedtraining signals in a frequency domain.